LMFDB - Embedded newform 414.3.b.a.91.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [414,3,Mod(91,414)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(414, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("414.91");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 414 = 2 \cdot 3^{2} \cdot 23 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	414.b (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$11.2806829445$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.613376.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 58x^{2} + 599 $ x^4 + 58*x^2 + 599
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 46)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			91.4
Root			$-6.67505i$ of defining polynomial
Character	$\chi$	$=$	414.91
Dual form			414.3.b.a.91.3

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.41421 q^{2} +2.00000 q^{4} +9.43995i q^{5} -3.91016i q^{7} +2.82843 q^{8} +O(q^{10})$	\(q+1.41421 q^{2} +2.00000 q^{4} +9.43995i q^{5} -3.91016i q^{7} +2.82843 q^{8} +13.3501i q^{10} -3.91016i q^{11} +10.3137 q^{13} -5.52980i q^{14} +4.00000 q^{16} +17.2603i q^{17} +26.7002i q^{19} +18.8799i q^{20} -5.52980i q^{22} +(-3.10051 + 22.7901i) q^{23} -64.1127 q^{25} +14.5858 q^{26} -7.82031i q^{28} -35.1421 q^{29} +33.2426 q^{31} +5.65685 q^{32} +24.4097i q^{34} +36.9117 q^{35} -39.3794i q^{37} +37.7598i q^{38} +26.7002i q^{40} -13.4853 q^{41} +29.9395i q^{43} -7.82031i q^{44} +(-4.38478 + 32.2300i) q^{46} +31.0416 q^{47} +33.7107 q^{49} -90.6690 q^{50} +20.6274 q^{52} -61.2207i q^{53} +36.9117 q^{55} -11.0596i q^{56} -49.6985 q^{58} +26.2010 q^{59} -15.6406i q^{61} +47.0122 q^{62} +8.00000 q^{64} +97.3609i q^{65} -68.3702i q^{67} +34.5205i q^{68} +52.2010 q^{70} +111.267 q^{71} +47.8528 q^{73} -55.6910i q^{74} +53.4004i q^{76} -15.2893 q^{77} -117.860i q^{79} +37.7598i q^{80} -19.0711 q^{82} -74.8487i q^{83} -162.936 q^{85} +42.3408i q^{86} -11.0596i q^{88} -39.1016i q^{89} -40.3282i q^{91} +(-6.20101 + 45.5801i) q^{92} +43.8995 q^{94} -252.049 q^{95} -66.0797i q^{97} +47.6741 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4}+O(q^{10}) $ 4 * q + 8 * q^4	$ 4 q + 8 q^{4} - 4 q^{13} + 16 q^{16} - 52 q^{23} - 132 q^{25} + 64 q^{26} - 84 q^{29} + 116 q^{31} - 56 q^{35} - 20 q^{41} + 56 q^{46} + 28 q^{47} - 148 q^{49} - 176 q^{50} - 8 q^{52} - 56 q^{55} - 80 q^{58} + 184 q^{59} + 24 q^{62} + 32 q^{64} + 288 q^{70} + 100 q^{71} - 148 q^{73} - 344 q^{77} - 48 q^{82} - 120 q^{85} - 104 q^{92} + 136 q^{94} - 352 q^{95} + 400 q^{98}+O(q^{100}) $ 4 * q + 8 * q^4 - 4 * q^13 + 16 * q^16 - 52 * q^23 - 132 * q^25 + 64 * q^26 - 84 * q^29 + 116 * q^31 - 56 * q^35 - 20 * q^41 + 56 * q^46 + 28 * q^47 - 148 * q^49 - 176 * q^50 - 8 * q^52 - 56 * q^55 - 80 * q^58 + 184 * q^59 + 24 * q^62 + 32 * q^64 + 288 * q^70 + 100 * q^71 - 148 * q^73 - 344 * q^77 - 48 * q^82 - 120 * q^85 - 104 * q^92 + 136 * q^94 - 352 * q^95 + 400 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/414\mathbb{Z}\right)^\times$.

$n$	$47$	$235$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$	1.41421			0.707107
$2$	1.41421			0.707107
$3$		0			0
$3$		0			0
$4$	2.00000			0.500000
$5$			9.43995i			1.88799i	0.329959	+	0.943995i	$0.392965\pi$
$5$			9.43995i			1.88799i	−0.329959	+	0.943995i	$0.607035\pi$
$6$		0			0
$7$		−	3.91016i		−	0.558594i	−0.960205	−	0.279297i	$-0.909899\pi$
$7$		−	3.91016i		−	0.558594i	0.960205	−	0.279297i	$-0.0901014\pi$
$8$	2.82843			0.353553
$9$		0			0
$10$			13.3501i			1.33501i
$11$		−	3.91016i		−	0.355469i	−0.984078	−	0.177734i	$-0.943123\pi$
$11$		−	3.91016i		−	0.355469i	0.984078	−	0.177734i	$-0.0568768\pi$
$12$		0			0
$13$	10.3137			0.793362			0.396681	−	0.917956i	$-0.370162\pi$
$13$	10.3137			0.793362			0.396681	+	0.917956i	$0.370162\pi$
$14$		−	5.52980i		−	0.394985i
$15$		0			0
$16$	4.00000			0.250000
$17$			17.2603i			1.01531i	0.861561	+	0.507655i	$0.169487\pi$
$17$			17.2603i			1.01531i	−0.861561	+	0.507655i	$0.830513\pi$
$18$		0			0
$19$			26.7002i			1.40527i	0.711548	+	0.702637i	$0.247994\pi$
$19$			26.7002i			1.40527i	−0.711548	+	0.702637i	$0.752006\pi$
$20$			18.8799i			0.943995i
$21$		0			0
$22$		−	5.52980i		−	0.251354i
$23$	−3.10051	+	22.7901i	−0.134805	+	0.990872i
$23$	−3.10051	+	22.7901i	−0.134805	+	0.990872i
$24$		0			0
$25$	−64.1127			−2.56451
$26$	14.5858			0.560992
$27$		0			0
$28$		−	7.82031i		−	0.279297i
$29$	−35.1421			−1.21180			−0.605899	−	0.795542i	$-0.707186\pi$
$29$	−35.1421			−1.21180			−0.605899	+	0.795542i	$0.707186\pi$
$30$		0			0
$31$	33.2426			1.07234			0.536172	−	0.844109i	$-0.319870\pi$
$31$	33.2426			1.07234			0.536172	+	0.844109i	$0.319870\pi$
$32$	5.65685			0.176777
$33$		0			0
$34$			24.4097i			0.717932i
$35$	36.9117			1.05462
$36$		0			0
$37$		−	39.3794i		−	1.06431i	−0.846647	−	0.532155i	$-0.821382\pi$
$37$		−	39.3794i		−	1.06431i	0.846647	−	0.532155i	$-0.178618\pi$
$38$			37.7598i			0.993679i
$39$		0			0
$40$			26.7002i			0.667505i
$41$	−13.4853			−0.328909			−0.164455	−	0.986385i	$-0.552586\pi$
$41$	−13.4853			−0.328909			−0.164455	+	0.986385i	$0.552586\pi$
$42$		0			0
$43$			29.9395i			0.696267i	0.937445	+	0.348134i	$0.113184\pi$
$43$			29.9395i			0.696267i	−0.937445	+	0.348134i	$0.886816\pi$
$44$		−	7.82031i		−	0.177734i
$45$		0			0
$46$	−4.38478	+	32.2300i	−0.0953212	+	0.700652i
$47$	31.0416			0.660460			0.330230	−	0.943900i	$-0.392874\pi$
$47$	31.0416			0.660460			0.330230	+	0.943900i	$0.392874\pi$
$48$		0			0
$49$	33.7107			0.687973
$50$	−90.6690			−1.81338
$51$		0			0
$52$	20.6274			0.396681
$53$		−	61.2207i		−	1.15511i	−0.816352	−	0.577554i	$-0.804007\pi$
$53$		−	61.2207i		−	1.15511i	0.816352	−	0.577554i	$-0.195993\pi$
$54$		0			0
$55$	36.9117			0.671122
$56$		−	11.0596i		−	0.197493i
$57$		0			0
$58$	−49.6985			−0.856870
$59$	26.2010			0.444085			0.222042	−	0.975037i	$-0.428728\pi$
$59$	26.2010			0.444085			0.222042	+	0.975037i	$0.428728\pi$
$60$		0			0
$61$		−	15.6406i		−	0.256404i	−0.991748	−	0.128202i	$-0.959079\pi$
$61$		−	15.6406i		−	0.256404i	0.991748	−	0.128202i	$-0.0409205\pi$
$62$	47.0122			0.758261
$63$		0			0
$64$	8.00000			0.125000
$65$			97.3609i			1.49786i
$66$		0			0
$67$		−	68.3702i		−	1.02045i	−0.860041	−	0.510225i	$-0.829562\pi$
$67$		−	68.3702i		−	1.02045i	0.860041	−	0.510225i	$-0.170438\pi$
$68$			34.5205i			0.507655i
$69$		0			0
$70$	52.2010			0.745729
$71$	111.267			1.56714			0.783571	−	0.621303i	$-0.213396\pi$
$71$	111.267			1.56714			0.783571	+	0.621303i	$0.213396\pi$
$72$		0			0
$73$	47.8528			0.655518			0.327759	−	0.944761i	$-0.393707\pi$
$73$	47.8528			0.655518			0.327759	+	0.944761i	$0.393707\pi$
$74$		−	55.6910i		−	0.752580i
$75$		0			0
$76$			53.4004i			0.702637i
$77$	−15.2893			−0.198563
$78$		0			0
$79$		−	117.860i		−	1.49190i	−0.665999	−	0.745952i	$-0.731995\pi$
$79$		−	117.860i		−	1.49190i	0.665999	−	0.745952i	$-0.268005\pi$
$80$			37.7598i			0.471998i
$81$		0			0
$82$	−19.0711			−0.232574
$83$		−	74.8487i		−	0.901792i	−0.892576	−	0.450896i	$-0.851105\pi$
$83$		−	74.8487i		−	0.901792i	0.892576	−	0.450896i	$-0.148895\pi$
$84$		0			0
$85$	−162.936			−1.91689
$86$			42.3408i			0.492335i
$87$		0			0
$88$		−	11.0596i		−	0.125677i
$89$		−	39.1016i		−	0.439343i	−0.975574	−	0.219672i	$-0.929501\pi$
$89$		−	39.1016i		−	0.439343i	0.975574	−	0.219672i	$-0.0704986\pi$
$90$		0			0
$91$		−	40.3282i		−	0.443167i
$92$	−6.20101	+	45.5801i	−0.0674023	+	0.495436i
$93$		0			0
$94$	43.8995			0.467016
$95$	−252.049			−2.65314
$96$		0			0
$97$		−	66.0797i		−	0.681234i	−0.940202	−	0.340617i	$-0.889364\pi$
$97$		−	66.0797i		−	0.681234i	0.940202	−	0.340617i	$-0.110636\pi$
$98$	47.6741			0.486470
$99$		0			0
$100$	−128.225			−1.28225
$101$	−101.338			−1.00335			−0.501674	−	0.865057i	$-0.667282\pi$
$101$	−101.338			−1.00335			−0.501674	+	0.865057i	$0.667282\pi$
$102$		0			0
$103$			101.549i			0.985912i	0.870054	+	0.492956i	$0.164084\pi$
$103$			101.549i			0.985912i	−0.870054	+	0.492956i	$0.835916\pi$
$104$	29.1716			0.280496
$105$		0			0
$106$		−	86.5792i		−	0.816785i
$107$			75.5196i			0.705791i	0.935663	+	0.352895i	$0.114803\pi$
$107$			75.5196i			0.705791i	−0.935663	+	0.352895i	$0.885197\pi$
$108$		0			0
$109$			138.638i			1.27191i	0.771727	+	0.635954i	$0.219393\pi$
$109$			138.638i			1.27191i	−0.771727	+	0.635954i	$0.780607\pi$
$110$	52.2010			0.474555
$111$		0			0
$112$		−	15.6406i		−	0.139648i
$113$		−	180.701i		−	1.59912i	−0.600585	−	0.799561i	$-0.705065\pi$
$113$		−	180.701i		−	1.59912i	0.600585	−	0.799561i	$-0.294935\pi$
$114$		0			0
$115$	−215.137	−	29.2686i	−1.87076	−	0.254510i
$116$	−70.2843			−0.605899
$117$		0			0
$118$	37.0538			0.314015
$119$	67.4903			0.567146
$120$		0			0
$121$	105.711			0.873642
$122$		−	22.1192i		−	0.181305i
$123$		0			0
$124$	66.4853			0.536172
$125$		−	369.222i		−	2.95378i
$126$		0			0
$127$	36.3898			0.286534			0.143267	−	0.989684i	$-0.454239\pi$
$127$	36.3898			0.286534			0.143267	+	0.989684i	$0.454239\pi$
$128$	11.3137			0.0883883
$129$		0			0
$130$			137.689i			1.05915i
$131$	65.3848			0.499120			0.249560	−	0.968359i	$-0.419714\pi$
$131$	65.3848			0.499120			0.249560	+	0.968359i	$0.419714\pi$
$132$		0			0
$133$	104.402			0.784978
$134$		−	96.6900i		−	0.721567i
$135$		0			0
$136$			48.8194i			0.358966i
$137$			65.8018i			0.480305i	0.970735	+	0.240152i	$0.0771974\pi$
$137$			65.8018i			0.480305i	−0.970735	+	0.240152i	$0.922803\pi$
$138$		0			0
$139$	29.4092			0.211577			0.105788	−	0.994389i	$-0.466263\pi$
$139$	29.4092			0.211577			0.105788	+	0.994389i	$0.466263\pi$
$140$	73.8234			0.527310
$141$		0			0
$142$	157.355			1.10814
$143$		−	40.3282i		−	0.282015i
$144$		0			0
$145$		−	331.740i		−	2.28786i
$146$	67.6741			0.463521
$147$		0			0
$148$		−	78.7589i		−	0.532155i
$149$			3.23928i			0.0217401i	0.999941	+	0.0108701i	$0.00346012\pi$
$149$			3.23928i			0.0217401i	−0.999941	+	0.0108701i	$0.996540\pi$
$150$		0			0
$151$	−2.41421			−0.0159882			−0.00799408	−	0.999968i	$-0.502545\pi$
$151$	−2.41421			−0.0159882			−0.00799408	+	0.999968i	$0.502545\pi$
$152$			75.5196i			0.496840i
$153$		0			0
$154$	−21.6224			−0.140405
$155$			313.809i			2.02457i
$156$		0			0
$157$			73.6221i			0.468931i	0.972124	+	0.234465i	$0.0753339\pi$
$157$			73.6221i			0.468931i	−0.972124	+	0.234465i	$0.924666\pi$
$158$		−	166.680i		−	1.05494i
$159$		0			0
$160$			53.4004i			0.333753i
$161$	89.1127	+	12.1235i	0.553495	+	0.0753010i
$162$		0			0
$163$	125.586			0.770465			0.385232	−	0.922820i	$-0.374121\pi$
$163$	125.586			0.770465			0.385232	+	0.922820i	$0.374121\pi$
$164$	−26.9706			−0.164455
$165$		0			0
$166$		−	105.852i		−	0.637663i
$167$	92.8284			0.555859			0.277929	−	0.960601i	$-0.410352\pi$
$167$	92.8284			0.555859			0.277929	+	0.960601i	$0.410352\pi$
$168$		0			0
$169$	−62.6274			−0.370576
$170$	−230.426			−1.35545
$171$		0			0
$172$			59.8790i			0.348134i
$173$	6.71068			0.0387900			0.0193950	−	0.999812i	$-0.493826\pi$
$173$	6.71068			0.0387900			0.0193950	+	0.999812i	$0.493826\pi$
$174$		0			0
$175$			250.691i			1.43252i
$176$		−	15.6406i		−	0.0888672i
$177$		0			0
$178$		−	55.2980i		−	0.310663i
$179$	−278.262			−1.55454			−0.777268	−	0.629170i	$-0.783395\pi$
$179$	−278.262			−1.55454			−0.777268	+	0.629170i	$0.783395\pi$
$180$		0			0
$181$			53.6783i			0.296565i	0.988945	+	0.148283i	$0.0473745\pi$
$181$			53.6783i			0.296565i	−0.988945	+	0.148283i	$0.952625\pi$
$182$		−	57.0327i		−	0.313367i
$183$		0			0
$184$	−8.76955	+	64.4600i	−0.0476606	+	0.350326i
$185$	371.740			2.00941
$186$		0			0
$187$	67.4903			0.360911
$188$	62.0833			0.330230
$189$		0			0
$190$	−356.451			−1.87606
$191$		−	168.692i		−	0.883207i	−0.897210	−	0.441603i	$-0.854410\pi$
$191$		−	168.692i		−	0.883207i	0.897210	−	0.441603i	$-0.145590\pi$
$192$		0			0
$193$	−95.3188			−0.493880			−0.246940	−	0.969031i	$-0.579425\pi$
$193$	−95.3188			−0.493880			−0.246940	+	0.969031i	$0.579425\pi$
$194$		−	93.4508i		−	0.481705i
$195$		0			0
$196$	67.4214			0.343987
$197$	−249.083			−1.26438			−0.632191	−	0.774813i	$-0.717844\pi$
$197$	−249.083			−1.26438			−0.632191	+	0.774813i	$0.717844\pi$
$198$		0			0
$199$			76.1905i			0.382867i	0.981506	+	0.191433i	$0.0613136\pi$
$199$			76.1905i			0.382867i	−0.981506	+	0.191433i	$0.938686\pi$
$200$	−181.338			−0.906690
$201$		0			0
$202$	−143.314			−0.709474
$203$			137.411i			0.676903i
$204$		0			0
$205$		−	127.300i		−	0.620978i
$206$			143.612i			0.697145i
$207$		0			0
$208$	41.2548			0.198341
$209$	104.402			0.499531
$210$		0			0
$211$	−71.1716			−0.337306			−0.168653	−	0.985675i	$-0.553942\pi$
$211$	−71.1716			−0.337306			−0.168653	+	0.985675i	$0.553942\pi$
$212$		−	122.441i		−	0.577554i
$213$		0			0
$214$			106.801i			0.499069i
$215$	−282.627			−1.31455
$216$		0			0
$217$		−	129.984i		−	0.599004i
$218$			196.064i			0.899374i
$219$		0			0
$220$	73.8234			0.335561
$221$			178.017i			0.805508i
$222$		0			0
$223$	−84.0833			−0.377055			−0.188527	−	0.982068i	$-0.560371\pi$
$223$	−84.0833			−0.377055			−0.188527	+	0.982068i	$0.560371\pi$
$224$		−	22.1192i		−	0.0987464i
$225$		0			0
$226$		−	255.550i		−	1.13075i
$227$		−	263.092i		−	1.15900i	−0.814974	−	0.579498i	$-0.803249\pi$
$227$		−	263.092i		−	1.15900i	0.814974	−	0.579498i	$-0.196751\pi$
$228$		0			0
$229$			141.321i			0.617124i	0.951204	+	0.308562i	$0.0998477\pi$
$229$			141.321i			0.617124i	−0.951204	+	0.308562i	$0.900152\pi$
$230$	−304.250	−	41.3921i	−1.32283	−	0.179966i
$231$		0			0
$232$	−99.3970			−0.428435
$233$	353.784			1.51839			0.759193	−	0.650866i	$-0.225594\pi$
$233$	353.784			1.51839			0.759193	+	0.650866i	$0.225594\pi$
$234$		0			0
$235$			293.032i			1.24694i
$236$	52.4020			0.222042
$237$		0			0
$238$	95.4457			0.401033
$239$	272.286			1.13927			0.569637	−	0.821897i	$-0.307084\pi$
$239$	272.286			1.13927			0.569637	+	0.821897i	$0.307084\pi$
$240$		0			0
$241$		−	443.678i		−	1.84099i	−0.390758	−	0.920493i	$-0.627787\pi$
$241$		−	443.678i		−	1.84099i	0.390758	−	0.920493i	$-0.372213\pi$
$242$	149.497			0.617758
$243$		0			0
$244$		−	31.2812i		−	0.128202i
$245$			318.227i			1.29889i
$246$		0			0
$247$			275.378i			1.11489i
$248$	94.0244			0.379131
$249$		0			0
$250$		−	522.159i		−	2.08864i
$251$		−	194.607i		−	0.775326i	−0.921801	−	0.387663i	$-0.873282\pi$
$251$		−	194.607i		−	0.775326i	0.921801	−	0.387663i	$-0.126718\pi$
$252$		0			0
$253$	89.1127	+	12.1235i	0.352224	+	0.0479188i
$254$	51.4630			0.202610
$255$		0			0
$256$	16.0000			0.0625000
$257$	22.1371			0.0861365			0.0430683	−	0.999072i	$-0.486287\pi$
$257$	22.1371			0.0861365			0.0430683	+	0.999072i	$0.486287\pi$
$258$		0			0
$259$	−153.980			−0.594517
$260$			194.722i			0.748930i
$261$		0			0
$262$	92.4680			0.352931
$263$			425.076i			1.61626i	0.589006	+	0.808129i	$0.299519\pi$
$263$			425.076i			1.61626i	−0.589006	+	0.808129i	$0.700481\pi$
$264$		0			0
$265$	577.921			2.18083
$266$	147.647			0.555063
$267$		0			0
$268$		−	136.740i		−	0.510225i
$269$	−411.215			−1.52868			−0.764341	−	0.644813i	$-0.776935\pi$
$269$	−411.215			−1.52868			−0.764341	+	0.644813i	$0.776935\pi$
$270$		0			0
$271$	−98.8183			−0.364643			−0.182322	−	0.983239i	$-0.558361\pi$
$271$	−98.8183			−0.364643			−0.182322	+	0.983239i	$0.558361\pi$
$272$			69.0411i			0.253827i
$273$		0			0
$274$			93.0578i			0.339627i
$275$			250.691i			0.911602i
$276$		0			0
$277$	−207.319			−0.748443			−0.374222	−	0.927339i	$-0.622090\pi$
$277$	−207.319			−0.748443			−0.374222	+	0.927339i	$0.622090\pi$
$278$	41.5908			0.149607
$279$		0			0
$280$	104.402			0.372864
$281$		−	31.0034i		−	0.110332i	−0.998477	−	0.0551661i	$-0.982431\pi$
$281$		−	31.0034i		−	0.110332i	0.998477	−	0.0551661i	$-0.0175688\pi$
$282$		0			0
$283$			225.888i			0.798191i	0.916909	+	0.399095i	$0.130676\pi$
$283$			225.888i			0.798191i	−0.916909	+	0.399095i	$0.869324\pi$
$284$	222.534			0.783571
$285$		0			0
$286$		−	57.0327i		−	0.199415i
$287$			52.7296i			0.183727i
$288$		0			0
$289$	−8.91674			−0.0308538
$290$		−	469.151i		−	1.61776i
$291$		0			0
$292$	95.7056			0.327759
$293$			162.099i			0.553238i	0.960980	+	0.276619i	$0.0892140\pi$
$293$			162.099i			0.553238i	−0.960980	+	0.276619i	$0.910786\pi$
$294$		0			0
$295$			247.336i			0.838428i
$296$		−	111.382i		−	0.376290i
$297$		0			0
$298$			4.58103i			0.0153726i
$299$	−31.9777	+	235.050i	−0.106949	+	0.786121i
$300$		0			0
$301$	117.068			0.388931
$302$	−3.41421			−0.0113053
$303$		0			0
$304$			106.801i			0.351319i
$305$	147.647			0.484088
$306$		0			0
$307$	342.299			1.11498			0.557490	−	0.830184i	$-0.311765\pi$
$307$	342.299			1.11498			0.557490	+	0.830184i	$0.311765\pi$
$308$	−30.5786			−0.0992813
$309$		0			0
$310$			443.793i			1.43159i
$311$	−464.654			−1.49406			−0.747032	−	0.664788i	$-0.768522\pi$
$311$	−464.654			−1.49406			−0.747032	+	0.664788i	$0.768522\pi$
$312$		0			0
$313$		−	95.1855i		−	0.304107i	−0.988372	−	0.152054i	$-0.951411\pi$
$313$		−	95.1855i		−	0.304107i	0.988372	−	0.152054i	$-0.0485886\pi$
$314$			104.117i			0.331584i
$315$		0			0
$316$		−	235.721i		−	0.745952i
$317$	210.142			0.662909			0.331454	−	0.943471i	$-0.392461\pi$
$317$	210.142			0.662909			0.331454	+	0.943471i	$0.392461\pi$
$318$		0			0
$319$			137.411i			0.430756i
$320$			75.5196i			0.235999i
$321$		0			0
$322$	126.024	+	17.1452i	0.391380	+	0.0532458i
$323$	−460.853			−1.42679
$324$		0			0
$325$	−661.240			−2.03458
$326$	177.605			0.544801
$327$		0			0
$328$	−38.1421			−0.116287
$329$		−	121.378i		−	0.368929i
$330$		0			0
$331$	66.2822			0.200248			0.100124	−	0.994975i	$-0.468076\pi$
$331$	66.2822			0.200248			0.100124	+	0.994975i	$0.468076\pi$
$332$		−	149.697i		−	0.450896i
$333$		0			0
$334$	131.279			0.393052
$335$	645.411			1.92660
$336$		0			0
$337$			453.348i			1.34525i	0.739985	+	0.672623i	$0.234832\pi$
$337$			453.348i			1.34525i	−0.739985	+	0.672623i	$0.765168\pi$
$338$	−88.5685			−0.262037
$339$		0			0
$340$	−325.872			−0.958447
$341$		−	129.984i		−	0.381185i
$342$		0			0
$343$		−	323.412i		−	0.942891i
$344$			84.6817i			0.246168i
$345$		0			0
$346$	9.49033			0.0274287
$347$	291.750			0.840779			0.420389	−	0.907344i	$-0.361893\pi$
$347$	291.750			0.840779			0.420389	+	0.907344i	$0.361893\pi$
$348$		0			0
$349$	150.951			0.432525			0.216263	−	0.976335i	$-0.430613\pi$
$349$	150.951			0.432525			0.216263	+	0.976335i	$0.430613\pi$
$350$			354.530i			1.01294i
$351$		0			0
$352$		−	22.1192i		−	0.0628386i
$353$	−377.059			−1.06816			−0.534078	−	0.845435i	$-0.679341\pi$
$353$	−377.059			−1.06816			−0.534078	+	0.845435i	$0.679341\pi$
$354$		0			0
$355$			1050.36i			2.95875i
$356$		−	78.2031i		−	0.219672i
$357$		0			0
$358$	−393.522			−1.09922
$359$			352.240i			0.981169i	0.871394	+	0.490584i	$0.163217\pi$
$359$			352.240i			0.981169i	−0.871394	+	0.490584i	$0.836783\pi$
$360$		0			0
$361$	−351.902			−0.974797
$362$			75.9126i			0.209703i
$363$		0			0
$364$		−	80.6564i		−	0.221584i
$365$			451.728i			1.23761i
$366$		0			0
$367$			320.843i			0.874232i	0.899405	+	0.437116i	$0.144000\pi$
$367$			320.843i			0.874232i	−0.899405	+	0.437116i	$0.856000\pi$
$368$	−12.4020	+	91.1602i	−0.0337011	+	0.247718i
$369$		0			0
$370$	525.720			1.42086
$371$	−239.383			−0.645236
$372$		0			0
$373$		−	156.128i		−	0.418575i	−0.977854	−	0.209287i	$-0.932886\pi$
$373$		−	156.128i		−	0.418575i	0.977854	−	0.209287i	$-0.0671144\pi$
$374$	95.4457			0.255203
$375$		0			0
$376$	87.7990			0.233508
$377$	−362.446			−0.961395
$378$		0			0
$379$		−	451.220i		−	1.19055i	−0.803520	−	0.595277i	$-0.797042\pi$
$379$		−	451.220i		−	1.19055i	0.803520	−	0.595277i	$-0.202958\pi$
$380$	−504.098			−1.32657
$381$		0			0
$382$		−	238.567i		−	0.624521i
$383$		−	423.178i		−	1.10490i	−0.833545	−	0.552452i	$-0.813692\pi$
$383$		−	423.178i		−	1.10490i	0.833545	−	0.552452i	$-0.186308\pi$
$384$		0			0
$385$		−	144.330i		−	0.374884i
$386$	−134.801			−0.349226
$387$		0			0
$388$		−	132.159i		−	0.340617i
$389$			11.0596i			0.0284308i	0.999899	+	0.0142154i	$0.00452506\pi$
$389$			11.0596i			0.0284308i	−0.999899	+	0.0142154i	$0.995475\pi$
$390$		0			0
$391$	−393.362	−	53.5155i	−1.00604	−	0.136868i
$392$	95.3482			0.243235
$393$		0			0
$394$	−352.257			−0.894053
$395$	1112.60			2.81670
$396$		0			0
$397$	−332.578			−0.837727			−0.418864	−	0.908049i	$-0.637571\pi$
$397$	−332.578			−0.837727			−0.418864	+	0.908049i	$0.637571\pi$
$398$			107.750i			0.270728i
$399$		0			0
$400$	−256.451			−0.641127
$401$			51.7808i			0.129129i	0.997914	+	0.0645646i	$0.0205658\pi$
$401$			51.7808i			0.129129i	−0.997914	+	0.0645646i	$0.979434\pi$
$402$		0			0
$403$	342.855			0.850757
$404$	−202.676			−0.501674
$405$		0			0
$406$			194.329i			0.478642i
$407$	−153.980			−0.378329
$408$		0			0
$409$	256.088			0.626133			0.313066	−	0.949731i	$-0.398644\pi$
$409$	256.088			0.626133			0.313066	+	0.949731i	$0.398644\pi$
$410$		−	180.030i		−	0.439097i
$411$		0			0
$412$			203.098i			0.492956i
$413$		−	102.450i		−	0.248063i
$414$		0			0
$415$	706.569			1.70257
$416$	58.3431			0.140248
$417$		0			0
$418$	147.647			0.353222
$419$			241.644i			0.576715i	0.957523	+	0.288358i	$0.0931092\pi$
$419$			241.644i			0.576715i	−0.957523	+	0.288358i	$0.906891\pi$
$420$		0			0
$421$		−	312.860i		−	0.743136i	−0.928406	−	0.371568i	$-0.878820\pi$
$421$		−	312.860i		−	0.743136i	0.928406	−	0.371568i	$-0.121180\pi$
$422$	−100.652			−0.238511
$423$		0			0
$424$		−	173.158i		−	0.408392i
$425$		−	1106.60i		−	2.60377i
$426$		0			0
$427$	−61.1573			−0.143225
$428$			151.039i			0.352895i
$429$		0			0
$430$	−399.696			−0.929524
$431$		−	672.412i		−	1.56012i	−0.625704	−	0.780060i	$-0.715188\pi$
$431$		−	672.412i		−	1.56012i	0.625704	−	0.780060i	$-0.284812\pi$
$432$		0			0
$433$			395.414i			0.913197i	0.889673	+	0.456598i	$0.150932\pi$
$433$			395.414i			0.913197i	−0.889673	+	0.456598i	$0.849068\pi$
$434$		−	183.825i		−	0.423560i
$435$		0			0
$436$			277.276i			0.635954i
$437$	−608.500	−	82.7842i	−1.39245	−	0.189437i
$438$		0			0
$439$	836.997			1.90660			0.953300	−	0.302026i	$-0.0976630\pi$
$439$	836.997			1.90660			0.953300	+	0.302026i	$0.0976630\pi$
$440$	104.402			0.237277
$441$		0			0
$442$			251.755i			0.569580i
$443$	227.905			0.514457			0.257229	−	0.966351i	$-0.417191\pi$
$443$	227.905			0.514457			0.257229	+	0.966351i	$0.417191\pi$
$444$		0			0
$445$	369.117			0.829476
$446$	−118.912			−0.266618
$447$		0			0
$448$		−	31.2812i		−	0.0698242i
$449$	439.019			0.977771			0.488886	−	0.872348i	$-0.337404\pi$
$449$	439.019			0.977771			0.488886	+	0.872348i	$0.337404\pi$
$450$		0			0
$451$			52.7296i			0.116917i
$452$		−	361.402i		−	0.799561i
$453$		0			0
$454$		−	372.068i		−	0.819534i
$455$	380.696			0.836695
$456$		0			0
$457$			40.1654i			0.0878893i	0.999034	+	0.0439447i	$0.0139925\pi$
$457$			40.1654i			0.0878893i	−0.999034	+	0.0439447i	$0.986007\pi$
$458$			199.859i			0.436373i
$459$		0			0
$460$	−430.274	−	58.5372i	−0.935379	−	0.127255i
$461$	−65.6762			−0.142465			−0.0712323	−	0.997460i	$-0.522693\pi$
$461$	−65.6762			−0.142465			−0.0712323	+	0.997460i	$0.522693\pi$
$462$		0			0
$463$	−673.436			−1.45450			−0.727252	−	0.686370i	$-0.759203\pi$
$463$	−673.436			−1.45450			−0.727252	+	0.686370i	$0.759203\pi$
$464$	−140.569			−0.302949
$465$		0			0
$466$	500.326			1.07366
$467$		−	310.685i		−	0.665278i	−0.943054	−	0.332639i	$-0.892061\pi$
$467$		−	310.685i		−	0.665278i	0.943054	−	0.332639i	$-0.107939\pi$
$468$		0			0
$469$	−267.338			−0.570017
$470$			414.409i			0.881722i
$471$		0			0
$472$	74.1076			0.157008
$473$	117.068			0.247501
$474$		0			0
$475$		−	1711.82i		−	3.60384i
$476$	134.981			0.283573
$477$		0			0
$478$	385.071			0.805588
$479$			230.469i			0.481146i	0.970631	+	0.240573i	$0.0773354\pi$
$479$			230.469i			0.481146i	−0.970631	+	0.240573i	$0.922665\pi$
$480$		0			0
$481$		−	406.148i		−	0.844383i
$482$		−	627.455i		−	1.30177i
$483$		0			0
$484$	211.421			0.436821
$485$	623.789			1.28616
$486$		0			0
$487$	372.360			0.764599			0.382299	−	0.924039i	$-0.375132\pi$
$487$	372.360			0.764599			0.382299	+	0.924039i	$0.375132\pi$
$488$		−	44.2384i		−	0.0906524i
$489$		0			0
$490$			450.041i			0.918451i
$491$	−258.026			−0.525512			−0.262756	−	0.964862i	$-0.584631\pi$
$491$	−258.026			−0.525512			−0.262756	+	0.964862i	$0.584631\pi$
$492$		0			0
$493$		−	606.563i		−	1.23035i
$494$			389.444i			0.788347i
$495$		0			0
$496$	132.971			0.268086
$497$		−	435.071i		−	0.875395i
$498$		0			0
$499$	−372.747			−0.746988			−0.373494	−	0.927632i	$-0.621840\pi$
$499$	−372.747			−0.746988			−0.373494	+	0.927632i	$0.621840\pi$
$500$		−	738.444i		−	1.47689i
$501$		0			0
$502$		−	275.215i		−	0.548238i
$503$			611.191i			1.21509i	0.794284	+	0.607546i	$0.207846\pi$
$503$			611.191i			1.21509i	−0.794284	+	0.607546i	$0.792154\pi$
$504$		0			0
$505$		−	956.627i		−	1.89431i
$506$	126.024	+	17.1452i	0.249060	+	0.0338837i
$507$		0			0
$508$	72.7797			0.143267
$509$	−357.985			−0.703310			−0.351655	−	0.936130i	$-0.614381\pi$
$509$	−357.985			−0.703310			−0.351655	+	0.936130i	$0.614381\pi$
$510$		0			0
$511$		−	187.112i		−	0.366168i
$512$	22.6274			0.0441942
$513$		0			0
$514$	31.3066			0.0609077
$515$	−958.617			−1.86139
$516$		0			0
$517$		−	121.378i		−	0.234773i
$518$	−217.760			−0.420387
$519$		0			0
$520$			275.378i			0.529574i
$521$		−	123.227i		−	0.236521i	−0.992983	−	0.118261i	$-0.962268\pi$
$521$		−	123.227i		−	0.236521i	0.992983	−	0.118261i	$-0.0377318\pi$
$522$		0			0
$523$			246.110i			0.470573i	0.971926	+	0.235286i	$0.0756028\pi$
$523$			246.110i			0.470573i	−0.971926	+	0.235286i	$0.924397\pi$
$524$	130.770			0.249560
$525$		0			0
$526$			601.148i			1.14287i
$527$			573.777i			1.08876i
$528$		0			0
$529$	−509.774	−	141.321i	−0.963655	−	0.267148i
$530$	817.304			1.54208
$531$		0			0
$532$	208.804			0.392489
$533$	−139.083			−0.260944
$534$		0			0
$535$	−712.902			−1.33253
$536$		−	193.380i		−	0.360784i
$537$		0			0
$538$	−581.546			−1.08094
$539$		−	131.814i		−	0.244553i
$540$		0			0
$541$	−425.922			−0.787286			−0.393643	−	0.919263i	$-0.628785\pi$
$541$	−425.922			−0.787286			−0.393643	+	0.919263i	$0.628785\pi$
$542$	−139.750			−0.257842
$543$		0			0
$544$			97.6388i			0.179483i
$545$	−1308.74			−2.40135
$546$		0			0
$547$	−715.070			−1.30726			−0.653629	−	0.756815i	$-0.726754\pi$
$547$	−715.070			−1.30726			−0.653629	+	0.756815i	$0.726754\pi$
$548$			131.604i			0.240152i
$549$		0			0
$550$			354.530i			0.644600i
$551$		−	938.303i		−	1.70291i
$552$		0			0
$553$	−460.853			−0.833369
$554$	−293.193			−0.529229
$555$		0			0
$556$	58.8183			0.105788
$557$		−	325.539i		−	0.584451i	−0.956349	−	0.292226i	$-0.905604\pi$
$557$		−	325.539i		−	0.584451i	0.956349	−	0.292226i	$-0.0943958\pi$
$558$		0			0
$559$			308.787i			0.552392i
$560$	147.647			0.263655
$561$		0			0
$562$		−	43.8454i		−	0.0780167i
$563$			636.780i			1.13105i	0.824732	+	0.565524i	$0.191326\pi$
$563$			636.780i			1.13105i	−0.824732	+	0.565524i	$0.808674\pi$
$564$		0			0
$565$	1705.81			3.01913
$566$			319.454i			0.564406i
$567$		0			0
$568$	314.711			0.554068
$569$		−	359.226i		−	0.631329i	−0.948871	−	0.315665i	$-0.897773\pi$
$569$		−	359.226i		−	0.631329i	0.948871	−	0.315665i	$-0.102227\pi$
$570$		0			0
$571$		−	766.371i		−	1.34216i	−0.741387	−	0.671078i	$-0.765832\pi$
$571$		−	766.371i		−	1.34216i	0.741387	−	0.671078i	$-0.234168\pi$
$572$		−	80.6564i		−	0.141008i
$573$		0			0
$574$			74.5709i			0.129914i
$575$	198.782	−	1461.13i	0.345707	−	2.54110i
$576$		0			0
$577$	−966.661			−1.67532			−0.837661	−	0.546190i	$-0.816078\pi$
$577$	−966.661			−1.67532			−0.837661	+	0.546190i	$0.816078\pi$
$578$	−12.6102			−0.0218169
$579$		0			0
$580$		−	663.480i		−	1.14393i
$581$	−292.670			−0.503735
$582$		0			0
$583$	−239.383			−0.410605
$584$	135.348			0.231761
$585$		0			0
$586$			229.242i			0.391199i
$587$	855.933			1.45815			0.729074	−	0.684435i	$-0.239951\pi$
$587$	855.933			1.45815			0.729074	+	0.684435i	$0.239951\pi$
$588$		0			0
$589$			887.586i			1.50694i
$590$			349.786i			0.592858i
$591$		0			0
$592$		−	157.518i		−	0.266077i
$593$	−159.563			−0.269078			−0.134539	−	0.990908i	$-0.542955\pi$
$593$	−159.563			−0.269078			−0.134539	+	0.990908i	$0.542955\pi$
$594$		0			0
$595$			637.106i			1.07077i
$596$			6.47856i			0.0108701i
$597$		0			0
$598$	−45.2233	+	332.411i	−0.0756243	+	0.555871i
$599$	46.1320			0.0770151			0.0385075	−	0.999258i	$-0.487740\pi$
$599$	46.1320			0.0770151			0.0385075	+	0.999258i	$0.487740\pi$
$600$		0			0
$601$	−283.010			−0.470899			−0.235449	−	0.971887i	$-0.575656\pi$
$601$	−283.010			−0.470899			−0.235449	+	0.971887i	$0.575656\pi$
$602$	165.559			0.275015
$603$		0			0
$604$	−4.82843			−0.00799408
$605$			997.904i			1.64943i
$606$		0			0
$607$	690.250			1.13715			0.568575	−	0.822632i	$-0.307495\pi$
$607$	690.250			1.13715			0.568575	+	0.822632i	$0.307495\pi$
$608$			151.039i			0.248420i
$609$		0			0
$610$	208.804			0.342302
$611$	320.154			0.523984
$612$		0			0
$613$			247.058i			0.403032i	0.979485	+	0.201516i	$0.0645867\pi$
$613$			247.058i			0.403032i	−0.979485	+	0.201516i	$0.935413\pi$
$614$	484.083			0.788409
$615$		0			0
$616$	−43.2447			−0.0702025
$617$		−	548.303i		−	0.888660i	−0.895863	−	0.444330i	$-0.853442\pi$
$617$		−	548.303i		−	0.888660i	0.895863	−	0.444330i	$-0.146558\pi$
$618$		0			0
$619$			1090.68i			1.76201i	0.473108	+	0.881005i	$0.343132\pi$
$619$			1090.68i			1.76201i	−0.473108	+	0.881005i	$0.656868\pi$
$620$			627.618i			1.01229i
$621$		0			0
$622$	−657.120			−1.05646
$623$	−152.893			−0.245414
$624$		0			0
$625$	1882.62			3.01219
$626$		−	134.613i		−	0.215036i
$627$		0			0
$628$			147.244i			0.234465i
$629$	679.700			1.08060
$630$		0			0
$631$		−	132.159i		−	0.209444i	−0.994502	−	0.104722i	$-0.966605\pi$
$631$		−	132.159i		−	0.209444i	0.994502	−	0.104722i	$-0.0333953\pi$
$632$		−	333.360i		−	0.527468i
$633$		0			0
$634$	297.186			0.468747
$635$			343.518i			0.540974i
$636$		0			0
$637$	347.682			0.545812
$638$			194.329i			0.304591i
$639$		0			0
$640$			106.801i			0.166876i
$641$			349.786i			0.545688i	0.962058	+	0.272844i	$0.0879644\pi$
$641$			349.786i			0.545688i	−0.962058	+	0.272844i	$0.912036\pi$
$642$		0			0
$643$		−	147.129i		−	0.228817i	−0.993434	−	0.114408i	$-0.963503\pi$
$643$		−	147.129i		−	0.228817i	0.993434	−	0.114408i	$-0.0364972\pi$
$644$	178.225	+	24.2469i	0.276748	+	0.0376505i
$645$		0			0
$646$	−651.744			−1.00889
$647$	−966.252			−1.49343			−0.746717	−	0.665142i	$-0.768371\pi$
$647$	−966.252			−1.49343			−0.746717	+	0.665142i	$0.768371\pi$
$648$		0			0
$649$		−	102.450i		−	0.157858i
$650$	−935.134			−1.43867
$651$		0			0
$652$	251.172			0.385232
$653$	817.378			1.25173			0.625863	−	0.779933i	$-0.284747\pi$
$653$	817.378			1.25173			0.625863	+	0.779933i	$0.284747\pi$
$654$		0			0
$655$			617.229i			0.942335i
$656$	−53.9411			−0.0822273
$657$		0			0
$658$		−	171.654i		−	0.260872i
$659$		−	1053.82i		−	1.59913i	−0.600581	−	0.799564i	$-0.705064\pi$
$659$		−	1053.82i		−	1.59913i	0.600581	−	0.799564i	$-0.294936\pi$
$660$		0			0
$661$		−	1162.41i		−	1.75856i	−0.476305	−	0.879280i	$-0.658024\pi$
$661$		−	1162.41i		−	1.75856i	0.476305	−	0.879280i	$-0.341976\pi$
$662$	93.7372			0.141597
$663$		0			0
$664$		−	211.704i		−	0.318832i
$665$			985.550i			1.48203i
$666$		0			0
$667$	108.958	−	800.891i	0.163356	−	1.20074i
$668$	185.657			0.277929
$669$		0			0
$670$	912.749			1.36231
$671$	−61.1573			−0.0911435
$672$		0			0
$673$	−647.461			−0.962052			−0.481026	−	0.876706i	$-0.659736\pi$
$673$	−647.461			−0.962052			−0.481026	+	0.876706i	$0.659736\pi$
$674$			641.131i			0.951233i
$675$		0			0
$676$	−125.255			−0.185288
$677$			544.278i			0.803956i	0.915649	+	0.401978i	$0.131677\pi$
$677$			544.278i			0.803956i	−0.915649	+	0.401978i	$0.868323\pi$
$678$		0			0
$679$	−258.382			−0.380533
$680$	−460.853			−0.677725
$681$		0			0
$682$		−	183.825i		−	0.269538i
$683$	−77.8314			−0.113955			−0.0569776	−	0.998375i	$-0.518146\pi$
$683$	−77.8314			−0.113955			−0.0569776	+	0.998375i	$0.518146\pi$
$684$		0			0
$685$	−621.166			−0.906811
$686$		−	457.373i		−	0.666725i
$687$		0			0
$688$			119.758i			0.174067i
$689$		−	631.413i		−	0.916419i
$690$		0			0
$691$	−223.769			−0.323833			−0.161917	−	0.986804i	$-0.551768\pi$
$691$	−223.769			−0.323833			−0.161917	+	0.986804i	$0.551768\pi$
$692$	13.4214			0.0193950
$693$		0			0
$694$	412.597			0.594520
$695$			277.621i			0.399455i
$696$		0			0
$697$		−	232.760i		−	0.333945i
$698$	213.477			0.305841
$699$		0			0
$700$			501.381i			0.716259i
$701$			376.995i			0.537795i	0.963169	+	0.268898i	$0.0866594\pi$
$701$			376.995i			0.537795i	−0.963169	+	0.268898i	$0.913341\pi$
$702$		0			0
$703$	1051.44			1.49565
$704$		−	31.2812i		−	0.0444336i
$705$		0			0
$706$	−533.242			−0.755300
$707$			396.248i			0.560464i
$708$		0			0
$709$			1182.12i			1.66731i	0.552287	+	0.833654i	$0.313755\pi$
$709$			1182.12i			1.66731i	−0.552287	+	0.833654i	$0.686245\pi$
$710$			1485.43i			2.09215i
$711$		0			0
$712$		−	110.596i		−	0.155331i
$713$	−103.069	+	757.602i	−0.144557	+	1.06256i
$714$		0			0
$715$	380.696			0.532443
$716$	−556.524			−0.777268
$717$		0			0
$718$			498.142i			0.693791i
$719$	−93.9941			−0.130729			−0.0653645	−	0.997861i	$-0.520821\pi$
$719$	−93.9941			−0.130729			−0.0653645	+	0.997861i	$0.520821\pi$
$720$		0			0
$721$	397.072			0.550724
$722$	−497.664			−0.689285
$723$		0			0
$724$			107.357i			0.148283i
$725$	2253.06			3.10767
$726$		0			0
$727$		−	1303.61i		−	1.79314i	−0.442900	−	0.896571i	$-0.646050\pi$
$727$		−	1303.61i		−	1.79314i	0.442900	−	0.896571i	$-0.353950\pi$
$728$		−	114.065i		−	0.156683i
$729$		0			0
$730$			638.840i			0.875124i
$731$	−516.764			−0.706927
$732$		0			0
$733$			635.946i			0.867594i	0.901011	+	0.433797i	$0.142826\pi$
$733$			635.946i			0.867594i	−0.901011	+	0.433797i	$0.857174\pi$
$734$			453.741i			0.618176i
$735$		0			0
$736$	−17.5391	+	128.920i	−0.0238303	+	0.175163i
$737$	−267.338			−0.362738
$738$		0			0
$739$	200.380			0.271150			0.135575	−	0.990767i	$-0.456712\pi$
$739$	200.380			0.271150			0.135575	+	0.990767i	$0.456712\pi$
$740$	743.480			1.00470
$741$		0			0
$742$	−338.538			−0.456251
$743$			1210.79i			1.62959i	0.579748	+	0.814796i	$0.303151\pi$
$743$			1210.79i			1.62959i	−0.579748	+	0.814796i	$0.696849\pi$
$744$		0			0
$745$	−30.5786			−0.0410452
$746$		−	220.799i		−	0.295977i
$747$		0			0
$748$	134.981			0.180455
$749$	295.294			0.394250
$750$		0			0
$751$			983.883i			1.31010i	0.755587	+	0.655048i	$0.227352\pi$
$751$			983.883i			1.31010i	−0.755587	+	0.655048i	$0.772648\pi$
$752$	124.167			0.165115
$753$		0			0
$754$	−512.576			−0.679809
$755$		−	22.7901i		−	0.0301855i
$756$		0			0
$757$			1494.15i			1.97378i	0.161407	+	0.986888i	$0.448397\pi$
$757$			1494.15i			1.97378i	−0.161407	+	0.986888i	$0.551603\pi$
$758$		−	638.122i		−	0.841849i
$759$		0			0
$760$	−712.902			−0.938028
$761$	779.132			1.02383			0.511913	−	0.859037i	$-0.328937\pi$
$761$	779.132			1.02383			0.511913	+	0.859037i	$0.328937\pi$
$762$		0			0
$763$	542.096			0.710479
$764$		−	337.385i		−	0.441603i
$765$		0			0
$766$		−	598.464i		−	0.781285i
$767$	270.230			0.352320
$768$		0			0
$769$			155.017i			0.201582i	0.994908	+	0.100791i	$0.0321374\pi$
$769$			155.017i			0.201582i	−0.994908	+	0.100791i	$0.967863\pi$
$770$		−	204.114i		−	0.265083i
$771$		0			0
$772$	−190.638			−0.246940
$773$			97.8690i			0.126609i	0.997994	+	0.0633047i	$0.0201640\pi$
$773$			97.8690i			0.126609i	−0.997994	+	0.0633047i	$0.979836\pi$
$774$		0			0
$775$	−2131.28			−2.75003
$776$		−	186.902i		−	0.240852i
$777$		0			0
$778$			15.6406i			0.0201036i
$779$		−	360.060i		−	0.462208i
$780$		0			0
$781$		−	435.071i		−	0.557070i
$782$	−556.299	−	75.6824i	−0.711379	−	0.0967806i
$783$		0			0
$784$	134.843			0.171993
$785$	−694.989			−0.885336
$786$		0			0
$787$		−	728.726i		−	0.925954i	−0.886370	−	0.462977i	$-0.846781\pi$
$787$		−	728.726i		−	0.925954i	0.886370	−	0.462977i	$-0.153219\pi$
$788$	−498.167			−0.632191
$789$		0			0
$790$	1573.45			1.99171
$791$	−706.569			−0.893260
$792$		0			0
$793$		−	161.313i		−	0.203421i
$794$	−470.336			−0.592363
$795$		0			0
$796$			152.381i			0.191433i
$797$		−	279.729i		−	0.350978i	−0.984481	−	0.175489i	$-0.943849\pi$
$797$		−	279.729i		−	0.350978i	0.984481	−	0.175489i	$-0.0561506\pi$
$798$		0			0
$799$			535.787i			0.670572i
$800$	−362.676			−0.453345
$801$		0			0
$802$			73.2291i			0.0913081i
$803$		−	187.112i		−	0.233016i
$804$		0			0
$805$	−114.445	+	841.220i	−0.142168	+	1.04499i
$806$	484.870			0.601576
$807$		0			0
$808$	−286.627			−0.354737
$809$	−1156.82			−1.42994			−0.714968	−	0.699157i	$-0.753559\pi$
$809$	−1156.82			−1.42994			−0.714968	+	0.699157i	$0.753559\pi$
$810$		0			0
$811$	1178.54			1.45319			0.726594	−	0.687067i	$-0.241102\pi$
$811$	1178.54			1.45319			0.726594	+	0.687067i	$0.241102\pi$
$812$			274.822i			0.338451i
$813$		0			0
$814$	−217.760			−0.267519
$815$			1185.52i			1.45463i
$816$		0			0
$817$	−799.391			−0.978447
$818$	362.164			0.442743
$819$		0			0
$820$		−	254.601i		−	0.310489i
$821$	732.347			0.892019			0.446009	−	0.895028i	$-0.352845\pi$
$821$	732.347			0.892019			0.446009	+	0.895028i	$0.352845\pi$
$822$		0			0
$823$	−882.012			−1.07170			−0.535852	−	0.844312i	$-0.680009\pi$
$823$	−882.012			−1.07170			−0.535852	+	0.844312i	$0.680009\pi$
$824$			287.224i			0.348573i
$825$		0			0
$826$		−	144.886i		−	0.175407i
$827$			1039.74i			1.25724i	0.777713	+	0.628619i	$0.216380\pi$
$827$			1039.74i			1.25724i	−0.777713	+	0.628619i	$0.783620\pi$
$828$		0			0
$829$	413.505			0.498799			0.249400	−	0.968401i	$-0.419767\pi$
$829$	413.505			0.498799			0.249400	+	0.968401i	$0.419767\pi$
$830$	999.239			1.20390
$831$		0			0
$832$	82.5097			0.0991703
$833$			581.855i			0.698506i
$834$		0			0
$835$			876.296i			1.04946i
$836$	208.804			0.249766
$837$		0			0
$838$			341.736i			0.407799i
$839$			254.486i			0.303320i	0.988433	+	0.151660i	$0.0484619\pi$
$839$			254.486i			0.303320i	−0.988433	+	0.151660i	$0.951538\pi$
$840$		0			0
$841$	393.970			0.468454
$842$		−	442.451i		−	0.525476i
$843$		0			0
$844$	−142.343			−0.168653
$845$		−	591.200i		−	0.699645i
$846$		0			0
$847$		−	413.345i		−	0.488011i
$848$		−	244.883i		−	0.288777i
$849$		0			0
$850$		−	1564.97i		−	1.84114i
$851$	897.460	+	122.096i	1.05459	+	0.143474i
$852$		0			0
$853$	744.132			0.872370			0.436185	−	0.899857i	$-0.356329\pi$
$853$	744.132			0.872370			0.436185	+	0.899857i	$0.356329\pi$
$854$	−86.4895			−0.101276
$855$		0			0
$856$			213.602i			0.249535i
$857$	578.921			0.675520			0.337760	−	0.941232i	$-0.390331\pi$
$857$	578.921			0.675520			0.337760	+	0.941232i	$0.390331\pi$
$858$		0			0
$859$	−1395.20			−1.62421			−0.812106	−	0.583510i	$-0.801679\pi$
$859$	−1395.20			−1.62421			−0.812106	+	0.583510i	$0.801679\pi$
$860$	−565.255			−0.657273
$861$		0			0
$862$		−	950.934i		−	1.10317i
$863$	−565.704			−0.655508			−0.327754	−	0.944763i	$-0.606292\pi$
$863$	−565.704			−0.655508			−0.327754	+	0.944763i	$0.606292\pi$
$864$		0			0
$865$			63.3485i			0.0732352i
$866$			559.200i			0.645728i
$867$		0			0
$868$		−	259.968i		−	0.299502i
$869$	−460.853			−0.530325
$870$		0			0
$871$		−	705.150i		−	0.809587i
$872$			392.127i			0.449687i
$873$		0			0
$874$	−860.548	−	117.074i	−0.984609	−	0.133952i
$875$	−1443.72			−1.64996
$876$		0			0
$877$	−468.112			−0.533765			−0.266882	−	0.963729i	$-0.585994\pi$
$877$	−468.112			−0.533765			−0.266882	+	0.963729i	$0.585994\pi$
$878$	1183.69			1.34817
$879$		0			0
$880$	147.647			0.167780
$881$		−	1317.52i		−	1.49548i	−0.663990	−	0.747741i	$-0.731138\pi$
$881$		−	1317.52i		−	1.49548i	0.663990	−	0.747741i	$-0.268862\pi$
$882$		0			0
$883$	−257.377			−0.291480			−0.145740	−	0.989323i	$-0.546556\pi$
$883$	−257.377			−0.291480			−0.145740	+	0.989323i	$0.546556\pi$
$884$			356.035i			0.402754i
$885$		0			0
$886$	322.306			0.363776
$887$	−331.722			−0.373982			−0.186991	−	0.982362i	$-0.559874\pi$
$887$	−331.722			−0.373982			−0.186991	+	0.982362i	$0.559874\pi$
$888$		0			0
$889$		−	142.290i		−	0.160056i
$890$	522.010			0.586528
$891$		0			0
$892$	−168.167			−0.188527
$893$			828.818i			0.928128i
$894$		0			0
$895$		−	2626.78i		−	2.93495i
$896$		−	44.2384i		−	0.0493732i
$897$		0			0
$898$	620.867			0.691389
$899$	−1168.22			−1.29946
$900$		0			0
$901$	1056.69			1.17279
$902$			74.5709i			0.0826728i
$903$		0			0
$904$		−	511.099i		−	0.565375i
$905$	−506.721			−0.559912
$906$		0			0
$907$		−	436.596i		−	0.481362i	−0.970604	−	0.240681i	$-0.922629\pi$
$907$		−	436.596i		−	0.481362i	0.970604	−	0.240681i	$-0.0773708\pi$
$908$		−	526.184i		−	0.579498i
$909$		0			0
$910$	538.386			0.591633
$911$		−	1530.29i		−	1.67979i	−0.542749	−	0.839895i	$-0.682617\pi$
$911$		−	1530.29i		−	1.67979i	0.542749	−	0.839895i	$-0.317383\pi$
$912$		0			0
$913$	−292.670			−0.320559
$914$			56.8025i			0.0621472i
$915$		0			0
$916$			282.643i			0.308562i
$917$		−	255.665i		−	0.278806i
$918$		0			0
$919$			676.783i			0.736434i	0.929740	+	0.368217i	$0.120032\pi$
$919$			676.783i			0.736434i	−0.929740	+	0.368217i	$0.879968\pi$
$920$	−608.500	−	82.7842i	−0.661413	−	0.0899828i
$921$		0			0
$922$	−92.8802			−0.100738
$923$	1147.58			1.24331
$924$		0			0
$925$			2524.72i			2.72943i
$926$	−952.382			−1.02849
$927$		0			0
$928$	−198.794			−0.214218
$929$	686.535			0.739004			0.369502	−	0.929230i	$-0.379528\pi$
$929$	686.535			0.739004			0.369502	+	0.929230i	$0.379528\pi$
$930$		0			0
$931$			900.082i			0.966791i
$932$	707.568			0.759193
$933$		0			0
$934$		−	439.375i		−	0.470422i
$935$			637.106i			0.681396i
$936$		0			0
$937$			774.354i			0.826418i	0.910636	+	0.413209i	$0.135592\pi$
$937$			774.354i			0.826418i	−0.910636	+	0.413209i	$0.864408\pi$
$938$	−378.073			−0.403063
$939$		0			0
$940$			586.063i			0.623471i
$941$		−	774.814i		−	0.823395i	−0.911321	−	0.411697i	$-0.864936\pi$
$941$		−	774.814i		−	0.823395i	0.911321	−	0.411697i	$-0.135064\pi$
$942$		0			0
$943$	41.8112	−	307.330i	0.0443385	−	0.325907i
$944$	104.804			0.111021
$945$		0			0
$946$	165.559			0.175010
$947$	1090.69			1.15173			0.575865	−	0.817545i	$-0.304665\pi$
$947$	1090.69			1.15173			0.575865	+	0.817545i	$0.304665\pi$
$948$		0			0
$949$	493.540			0.520063
$950$		−	2420.88i		−	2.54830i
$951$		0			0
$952$	190.891			0.200516
$953$		−	60.7603i		−	0.0637569i	−0.999492	−	0.0318785i	$-0.989851\pi$
$953$		−	60.7603i		−	0.0637569i	0.999492	−	0.0318785i	$-0.0101489\pi$
$954$		0			0
$955$	1592.45			1.66749
$956$	544.573			0.569637
$957$		0			0
$958$			325.932i			0.340222i
$959$	257.295			0.268295
$960$		0			0
$961$	144.073			0.149920
$962$		−	574.380i		−	0.597069i
$963$		0			0
$964$		−	887.356i		−	0.920493i
$965$		−	899.805i		−	0.932440i
$966$		0			0
$967$	1715.83			1.77438			0.887192	−	0.461400i	$-0.152653\pi$
$967$	1715.83			1.77438			0.887192	+	0.461400i	$0.152653\pi$
$968$	298.995			0.308879
$969$		0			0
$970$	882.171			0.909454
$971$		−	1071.57i		−	1.10358i	−0.833984	−	0.551789i	$-0.813946\pi$
$971$		−	1071.57i		−	1.10358i	0.833984	−	0.551789i	$-0.186054\pi$
$972$		0			0
$973$		−	114.994i		−	0.118185i
$974$	526.596			0.540653
$975$		0			0
$976$		−	62.5625i		−	0.0641009i
$977$			456.079i			0.466816i	0.972379	+	0.233408i	$0.0749877\pi$
$977$			456.079i			0.466816i	−0.972379	+	0.233408i	$0.925012\pi$
$978$		0			0
$979$	−152.893			−0.156173
$980$			636.454i			0.649443i
$981$		0			0
$982$	−364.905			−0.371593
$983$		−	216.055i		−	0.219791i	−0.993943	−	0.109896i	$-0.964948\pi$
$983$		−	216.055i		−	0.219791i	0.993943	−	0.109896i	$-0.0350517\pi$
$984$		0			0
$985$		−	2351.33i		−	2.38714i
$986$		−	857.809i		−	0.869989i
$987$		0			0
$988$			550.757i			0.557446i
$989$	−682.323	−	92.8276i	−0.689912	−	0.0938600i
$990$		0			0
$991$	−520.142			−0.524866			−0.262433	−	0.964950i	$-0.584525\pi$
$991$	−520.142			−0.524866			−0.262433	+	0.964950i	$0.584525\pi$
$992$	188.049			0.189565
$993$		0			0
$994$		−	615.284i		−	0.618998i
$995$	−719.235			−0.722849
$996$		0			0
$997$	466.877			0.468282			0.234141	−	0.972203i	$-0.424772\pi$
$997$	466.877			0.468282			0.234141	+	0.972203i	$0.424772\pi$
$998$	−527.144			−0.528201
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	414.3.b.a.91.4		4
3.2	odd	2		46.3.b.a.45.1	✓	4
12.11	even	2		368.3.f.c.321.3		4
15.2	even	4		1150.3.c.a.1149.4		8
15.8	even	4		1150.3.c.a.1149.5		8
15.14	odd	2		1150.3.d.a.551.4		4
23.22	odd	2	inner	414.3.b.a.91.3		4
24.5	odd	2		1472.3.f.f.321.4		4
24.11	even	2		1472.3.f.c.321.2		4
69.68	even	2		46.3.b.a.45.2	yes	4
276.275	odd	2		368.3.f.c.321.4		4
345.68	odd	4		1150.3.c.a.1149.6		8
345.137	odd	4		1150.3.c.a.1149.3		8
345.344	even	2		1150.3.d.a.551.3		4
552.275	odd	2		1472.3.f.c.321.1		4
552.413	even	2		1472.3.f.f.321.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
46.3.b.a.45.1	✓	4	3.2	odd	2
46.3.b.a.45.2	yes	4	69.68	even	2
368.3.f.c.321.3		4	12.11	even	2
368.3.f.c.321.4		4	276.275	odd	2
414.3.b.a.91.3		4	23.22	odd	2	inner
414.3.b.a.91.4		4	1.1	even	1	trivial
1150.3.c.a.1149.3		8	345.137	odd	4
1150.3.c.a.1149.4		8	15.2	even	4
1150.3.c.a.1149.5		8	15.8	even	4
1150.3.c.a.1149.6		8	345.68	odd	4
1150.3.d.a.551.3		4	345.344	even	2
1150.3.d.a.551.4		4	15.14	odd	2
1472.3.f.c.321.1		4	552.275	odd	2
1472.3.f.c.321.2		4	24.11	even	2
1472.3.f.f.321.3		4	552.413	even	2
1472.3.f.f.321.4		4	24.5	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 414 = 2 \cdot 3^{2} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	414.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+1.41421 q^{2} +2.00000 q^{4} +9.43995i q^{5} -3.91016i q^{7} +2.82843 q^{8} +O(q^{10})\)	\(q+1.41421 q^{2} +2.00000 q^{4} +9.43995i q^{5} -3.91016i q^{7} +2.82843 q^{8} +13.3501i q^{10} -3.91016i q^{11} +10.3137 q^{13} -5.52980i q^{14} +4.00000 q^{16} +17.2603i q^{17} +26.7002i q^{19} +18.8799i q^{20} -5.52980i q^{22} +(-3.10051 + 22.7901i) q^{23} -64.1127 q^{25} +14.5858 q^{26} -7.82031i q^{28} -35.1421 q^{29} +33.2426 q^{31} +5.65685 q^{32} +24.4097i q^{34} +36.9117 q^{35} -39.3794i q^{37} +37.7598i q^{38} +26.7002i q^{40} -13.4853 q^{41} +29.9395i q^{43} -7.82031i q^{44} +(-4.38478 + 32.2300i) q^{46} +31.0416 q^{47} +33.7107 q^{49} -90.6690 q^{50} +20.6274 q^{52} -61.2207i q^{53} +36.9117 q^{55} -11.0596i q^{56} -49.6985 q^{58} +26.2010 q^{59} -15.6406i q^{61} +47.0122 q^{62} +8.00000 q^{64} +97.3609i q^{65} -68.3702i q^{67} +34.5205i q^{68} +52.2010 q^{70} +111.267 q^{71} +47.8528 q^{73} -55.6910i q^{74} +53.4004i q^{76} -15.2893 q^{77} -117.860i q^{79} +37.7598i q^{80} -19.0711 q^{82} -74.8487i q^{83} -162.936 q^{85} +42.3408i q^{86} -11.0596i q^{88} -39.1016i q^{89} -40.3282i q^{91} +(-6.20101 + 45.5801i) q^{92} +43.8995 q^{94} -252.049 q^{95} -66.0797i q^{97} +47.6741 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4}+O(q^{10}) \) 4 * q + 8 * q^4	\( 4 q + 8 q^{4} - 4 q^{13} + 16 q^{16} - 52 q^{23} - 132 q^{25} + 64 q^{26} - 84 q^{29} + 116 q^{31} - 56 q^{35} - 20 q^{41} + 56 q^{46} + 28 q^{47} - 148 q^{49} - 176 q^{50} - 8 q^{52} - 56 q^{55} - 80 q^{58} + 184 q^{59} + 24 q^{62} + 32 q^{64} + 288 q^{70} + 100 q^{71} - 148 q^{73} - 344 q^{77} - 48 q^{82} - 120 q^{85} - 104 q^{92} + 136 q^{94} - 352 q^{95} + 400 q^{98}+O(q^{100}) \) 4 * q + 8 * q^4 - 4 * q^13 + 16 * q^16 - 52 * q^23 - 132 * q^25 + 64 * q^26 - 84 * q^29 + 116 * q^31 - 56 * q^35 - 20 * q^41 + 56 * q^46 + 28 * q^47 - 148 * q^49 - 176 * q^50 - 8 * q^52 - 56 * q^55 - 80 * q^58 + 184 * q^59 + 24 * q^62 + 32 * q^64 + 288 * q^70 + 100 * q^71 - 148 * q^73 - 344 * q^77 - 48 * q^82 - 120 * q^85 - 104 * q^92 + 136 * q^94 - 352 * q^95 + 400 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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